Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Trigonometry
Topic: The Six Trigonometric Functions
Subtopic: Basic Identities

Overview

Now that we have defined the six trig functions, we can begin to find relationships between them. These relationships are called identities. We will use these identities to simplify trigonometric expressions and to create even more identities. Simplifying a trig expression can be accomplished in many ways but the goal is to get it as simplified as possible.

The major techniques used to simplify a trig expression are:

• Convert to sines and cosines
• Use LCDs to combine fractions
• Rewrite expressions by substituting with known identities
• Perform an algebraic manipulations such as distributive law, FOIL, factor, simplify compound fractions

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 1.4.1 Use proper mathematical notation to write trig functions.
• 1.4.2 Given a trig ratio, information about the quadrant in which the angle lies, be able to sketch the triangle, label it properly, and find another trig ratio. For example, given tanθ=7/24, cosθ<0, find secθ.
• 1.4.3 Use the reciprocal identities, ratio identities, and Pythagorean identities to simplify trigonometric expressions.

Terminology

Memorize the 3 reciprocal identities, the 2 ratio identities, and at least the first of the 3 Pythagorean identities. These are too important and used too often to just have on paper. Commit them to your memory!

 Basic Trigonometric Identities to MEMORIZE Reciprocal Identities csc theta = 1/sintheta sec theta = 1/costheta cot theta = 1/tantheta Ratio (a.k.a. Quotient) Identities tantheta = sintheta/costheta cottheta = costheta/sintheta Pythagorean Identities cos^2theta+sin^2theta=1 1+tan^2theta=sec^2theta cot^2theta+1=csc^2theta

This diagram is a good way to visualize the three Pythagorean Identities. For instance by the Pythagorean Theorem on the middle triangle, do you see that 1 + (tantheta)^2 = (sectheta)^2?

Mini-Lectures and Examples

STUDY: Basic Identities

Supplemental Resources (optional)

Videos from James Sousa's MathIsPower4U:
Mini-Lesson: Trig Identities:  Reciprocal, Quotient, Pythagorean
Mini-Lesson: Fundamental Identities:  Reciprocal, Quotient, Pythagorean

rev. 2021-04-05